Localizing Vapor-Emitting Sources Using
With the ever increasing technological advances in networking and the miniaturization of electromechanical systems, the ability to deploy large groups of autonomous vehicles to perform various coordinated tasks is increasing. Some of the foreseeable tasks include search and recovery missions, hazardous waste clean up, exploration, and surveillance. Many of these tasks cannot be performed by a single agent. Furthermore, having many vehicles provides robustness to failures of a single agent, or communication links.
In particular, we would like to devise a method to estimate the location of a vapor-emitting source using sensor mounted mobile robots that communicate through a distributed network. Such a method would be useful in applications such as explosives detection, locating hazardous chemical leaks, and pollution sensing.
Figure 1: Depiction of sensor mounted robots measuring vapor concentration
To model the concentration of a diffusing substance at a point , we use the solution to the classical diffusion equation for a source free volume and constant diffusivity:
where is the complimentary error function. The concentration depends on the location of the source , the diffusivity in m2/s, the diffusion rate in Kg/s, and the initial time of diffusion . These are all unknown parameters.
The concentration measurements will invariably be affected by the presence for foreign materials, and sensor noise. Therefore we may model the sensor measurements in the following fashion:
where is the bias term representing the sensor’s response to foreign substances, and is the sensor’s noise. We may assume the noise is Gaussian distributed with zero mean and variance .
Taking measurements at points at different times, we may collect the measurements into a vector form
where , and . The vector is an -dimensional vector whose -th component is , , . The same can be said for the vector . Finally, is an matrix whose -th row is
To obtain an estimate of the unknown parameters, maximum likelihood estimation is used. To obtain a measure of the variability of these estimates the Cramer-Rao bound (CRB) in computed.
Communication between the mobile agents is conducted in a distributed fashion. Each agent has the capability to communicate only with other agents in its communication range. This is due in part to the spatially-distributed nature and limited communication capabilities of a mobile network. Also, we would like to reduce the computational complexity of our sensing algorithm. Using a distributed network the computational complexity grows linearly as opposed to quadratically in a centralized communication scheme.
The Moving-Sensor Algorithm
Our tactic for moving the mobile agents is as follows: Suppose we have already taken measurements with a time interval between measurements. We want to place the agents in locations where the CRB is a minimum. To accomplish this, each agent computes the gradient of the CRB given its measurements up to and including and then moves in the direction opposite to the gradient to a point that it can reach at time . We continue this tactic, and estimating the unknown parameters at each new location, until the CRB computed by one of the agents goes below some pre-defined threshold.
At this current time, we have successfully simulated the moving-sensor algorithm for one mobile agent, and for a distributed network of mobile agents. We have also begun analyzing a linearized version of the sensor model. Here is a sample of our results for the case of one mobile agent and multiple mobile agents.
One Sensor Algorithm
In this simulation, we have placed the vapor source (denoted by the “x”) at the coordinates (-50, -50). Sensing begins at 100 s after the beginning of emission. The sensor first collects concentration data by traversing a circle centered at (0, 0) and having a radius of 250 m. Data is collected every 10 s. At time = 400 s the moving-sensor algorithm begins using the data accumulated during the first 400 s. After 22 more measurements (220 s) the CRB of both the x and y coordinates of the vapor source go below 0.1 and the algorithm stops. As can be seen from the plot of the estimates of x and y that they converge to around (-50, -50).
Multiple Sensor Algorithm
In this multiple sensor simulation, the vapor source (denote by the red circle) is again placed at the coordinates (-50, -50). Again sensing begins 100 s after the beginning of emission. The sensors (denoted by the “x”’s) collect concentration data for 300 s and share this data with their neighbors. Each agent has a communication radius of 300 m. At time = 400 s the algorithm begins, after 6 more measurements one of the agents measures the CRB of the x and y coordinates of the vapor source to go below 0.1 and the algorithm stops. The CRB of the two coordinates as measured by the algorithm terminating agent are shown in the upper right hand corner. The plot on lower left shows the same agent’s estimates of the x and y coordinates of the vapor source.
We hope to have a full understanding of the convergence properties of the moving-sensor algorithm, and have designed an optimum distributed communication network.
B. Porat and A. Nehorai. “Localizing vapor-emitting sources by moving sensors,” IEEE Trans. Signal Processing, vol. 44, no. 2, pp.1018-1021, Apr. 1996.